Positive Borel measures

This is a note of real and complex analysis chapter 2.

Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set ($$C$$) ($$\Lambda f$$) represents the integration of the function ($$\int f du$$) (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let $$\Lambda$$ be a positive linear functional on $$C_c(X)$$. Then there exist a $$\sigma-algebra$$ in $$X$$ which contains all Borel sets in $$X$$, and there exists a unique positive measure $$mu$$ on $$\mathfrak{M}$$ which represents $$\Lambda$$ in the sense that (a) $$\Lambda f = \int f d \mu$$ for every $$f \in C_c(X)$$ and following additional properties:
(b) $$\mu(K) < \infty$$ for every compact set $$K \subset X$$.
(c) For every $$E \in \mathfrak{M}$$, we have $\mu(E) = inf\{\mu(V): E in V, V open\}$.
(d) The relation $\mu(E)=sup\{\mu(K): K \in E, K compact\}$
holds for every open set $$E$$, and for every $$E \in M$$ with $$\mu(E) < \infty$$.
(e) If $$E \in \mathfrak{M}, A subset E$$, and $$\mu(E) = 0$$, then $$A \in \mathfrak{M}$$.

The Riesz theorem is about linear functional $$\Lambda$$ is equivalently replaced with choosing measure $$\mu(E)=sup\{\Lambda f: f \prec V\}$$. Note $$sup \{\int^1_0 f(x)dx = \Lambda f: f \prec V, V (0,1) \} = 1$$. The notion of $$\prec$$ include $$0 \le f \le 1$$.

I confused $$C_c(X)$$ Jun Kang
Clinical Assistant Professor of Hospital Pathology

My research interests include pathology, oncology and statistics.